Athermal laser using a one-eighth wave faraday rotator

ABSTRACT

A laser device with a laser rod positioned within an optical resonant cavity. The composition of the rod is chosen to have a coefficient of linear expansion Alpha L and a thermal coefficient of the index of refraction Alpha n such that the average of the optical path lengths for radial and tangential polarization is equal to the optical path lengths for a ray through the center of the rod. A mode selecting aperture is positioned at an end of the rod along the axis of the cavity for discriminating against all but the HE11 mode. A 1/8 wave Faraday rotator is also positioned along the axis of the cavity. The result is a laser device in which the thermal gradient effects are substantially eliminated.

United States Patent [72] Inventor Elias Snitzer Wellesley, Mass.

[21] Appl. No. 817,867

[22] Filed Apr. 21, 1969 [45] Patented Dec. 21, 1971 [7 3] AssigneeAmerican Optical Corporation Southbridge, Mass.

Continuation-impart of application Ser. No.

375,568, June 16, 1964, now abandoned. This application Apr. 21, 1969,Ser. No. 817,867

[54] ATHERMAL LASER USING A ONE-EIGHTH WAVE Primary Examiner-Ronald L.Wibert Assistant Examiner-Edward S. Bauer Attorneys-William C. Nealon,Noble S. Williams, Robert Bird and Bernard L. Sweeney ABSTRACT: A laserdevice with a laser rod positioned within an optical resonant cavity.The composition of the rod is chosen to have a coefficient of linearexpansion 01,, and a thermal coefficient of the index of refraction or,such that the average of the optical path lengths for radial andtangential polarization is equal to the optical path lengths for a raythrough the center of the rod. A mode: selecting aperture is positionedat an end of the rod along the axis of the cavity for discriminatingagainst all but the HE mode. A Vs wave Faraday rotator is alsopositioned along the axis of the cavity. The result is a laser device inwhich the thermal gradient effects are substantially eliminated.

PATENTEU nEc2I I971 3.629323 SHEET 2 [1F 4 [AXIS IPERIPHERY TENSION A[RADIAL I m "mi 8 Z E I T 32 U] COMPRESSION TANGENTIAL.

STRESS DISTRIBUTION FOR AVERAGE OF RADIALLY AND TANGENTIALLY POLARIZEDLIGHT DIRECTIONS OF PROPAGATION IaI 'S I07 H8 I IIO T w I08 NOTES H4 I20I I08 (VARIATION OFWAVE FRONT DUE To STRESS FOR TANGENTIALLY POLARIZEDLIGHT I20 (VARIATION OF WAVE FRONT DUE To STRESS FOR AVERAGE OF RADIALLYAND TANOENTIALLY POLARIZED LIGHT) H4 (VARIATION OFvVAVE FRONT DUE TosTREss FOR RADIALLY POLARIZED LIGHT) H6 (VA IATION OF WAVE FRONT DUE ToTEMPERATURE INVENTOR. ELI/R8 SNITZE R PATENTED nc21 I971 3529,7213

sum 3 OF 4 AVERAGE OF THE LONG ROD 8 FOR TANGENTIAL AND [RADIALPOLARlZATION /--TA NGENTIAL b LO \38 POLARIZATION 0.9 .88 1 -RAD|AL 08AVERAGE OF 2/ FOR RADIAL AND poLARllAT'oN 0.7 TANGENTIAL POLARIZATION o6 AND FOR EXTREME CONFIGURKTIONS d1 am JNVENTOR.

ELIAS SNITZER 4'15 7 V SHEET 4 UF 4 TI T oNciENTRIc CYLINDER 67x10 mmLONG ROD (RADIAL I2 x I0" 0. a?

PoLARIzATIoN) LONG ROD (TA NGEN'HAL 34 X IO 0.99

POLARIZATION) LONG ROD(AVERA6E OF RADIALi TANGEN'HAL 38 x IO 0. 93

POLARIZATION) STACKED DISC ADIAL -48 |o 0.79

POLARIZATION) STACKED DIsc (TANGENTIAL POLARIZATION) 42 x 87 STACKEDoIsc (AVERAGEOF RADIAL i 5 I0 0. 83 TANGENTIAL POLARIZATION) INVENTORELIAS SNITZER AT'T R y ATHERMAL LASER USING A ONE-EIGHTH WAVE FARADAYROTATOR CROSS REFERENCES TO RELATED APPLICATIONS This application is acontinuation-in-part of application Ser. No. 375,568 filed June 16, 1964now abandoned.

This invention relates to laser apparatus and more particularly to thecomposition of laser devices and apparatus associated therewith toenable the elimination of the effects of the thermal gradients in theresonant cavity.

Lasers, otherwise referred to as optical masers, are lightamplifying orlight-producing devices and are specifically adapted to provide anoutput of high-intensity, coherent, monochromatic light. Such light isproduced in a laser (an acronym for light amplification by stimulatedemission of radiation) by photonic emission from the active atoms of abody or core composed of a so-called laser material. These atoms, whichare in a positive temperature state absorb a quantum of light from aflash lamp, said light being at a frequency proportional to thedifference in energy between two of the energy levels of the atom. Theatoms are, therefore, pumped or excited to a high energy level and anegative temperature state of population inversion, from which theyrapidly relax to a more stable intermediate level (still above theoriginal level). The atoms then relax, at a somewhat slower rate, fromthis intermediate level to the original level with an attendant emissionof light. This emission by the spontaneous relaxation to the originallevel is fluorescence, which stimulates a further relaxation of atomsstill at the intermediate level and the desired laser output isobtained. The laser output is coherent since it moves in phase with thestimulating fluorescent light given off from the spontaneous emission,and intensity in a narrow cone is provided by the structure of a laser,which is in the form of a rod, one end of which is totally reflectiveand the other end of which is slightly transmissive. The only lightwhich is strongly amplified is that which moves in the same direction asthe fluorescent light (perpendicular to the ends) and, therefore, theoutput has almost all of its intensity in a narrow cone.

By way of example, one conventional form of laser structure includes arod-shaped body composed of a suitable solid laser host materialcontaining a quantity of activator material, said body being surroundedconcentrically by a helical gaseousdischarge flash tube adapted to emita pulse of light specifically including light in the wavelength of anabsorption band of the laser material. When the flash tube is actuated,this light pulse enters the laser body, is absorbed by the lasermaterial, and thereby pumps the body with energy of such absorptivewavelength. This pumping excites active atoms in the laser body to shiftfrom an initial low energy level, in a series of interlevel transitionstypically involving a first energy-absorptive transition, to ashort-lived high energy level and an immediately subsequent spontaneousnonradiative transition (with release of heat energy but presentlyregarded as nonemissive) from this unstable level to the somewhat morestable upper energy level referred to above (intermediate in energybetween the aforementioned initial and unstable levels), and from whichlight-emissive transition occurs. Thus, the pumping pulse provides theexcitation step in laser operation, creating a very large population ofatoms at the upper energy level in the laser body. The establishment ofthis large upper level population is referred to as an inversion ofenergy states of the body.

For effecting induced light-emissive transition from this level tocomplete the atomic cycle of laser operation, the laser body of thestructure is disposed coaxially within a resonant cavity between opposedreflective cavity ends. Immediately upon the inversion of energy statesof the body, individual atoms at the aforementioned upper energy levelbegin to undergo emissive transition, spontaneously shifting to a lowerenergy level or terminal level (which may or may not be the initial,lowest energy level, i.e. the ground state, depending on the nature ofthe laser material used) with concomitant emission of light. Since thisupper energy level is relatively stable in a laser material, suchspontaneous emission would deplete the enlarged upper level populationat a comparatively slow rate. However, a portion of the light emitted bythe spontaneously emitting atoms passes through the resonant cavity tothe ends thereof and is thence reflected back-and-forth through thecavity between the reflective cavity ends, passing and repassing inmultiple bidirectional reflections. This bidirectionally reflected lightimmediately excites other atoms at the upper energy level to induce themto undergo emissive transition to the terminal level, producing morelight, which augments the bidirectionally reflected light in the cavityto induce still further emissive transitions from the upper levelpopulation. In such fashion a rising pulse of bidirectionally reflectedlight quickly develops within the cavity, reaching a quantitativelylarge value as the induced emissive transition of atoms from the upperlevel population becomes massive. Light of high intensity is accordinglycreated in one or a succession of light pulses while the pumping lightis present, the action continuing until depletion of this population bysuch transitions restores the laser body to a normal energy state. Topermit emission of such portion of this large bidirectionally reflectedlight pulse or pulses from the laser cavity, one reflective end of thecavity is made partially transmissive. The fraction of thebidirectionally reflected light escaping therethrough constitutes thelaser output pulse.

It has been found that the intensity of the useful portion of the laseroutput pulse can be enhanced by restricting the bidirectional reflectionof light in the laser cavity to light emitted in certain selected modesof propagation. The atoms in a laser body emit light in a plurality ofsuch modes, including the modes for the plane waves propagated parallelto the long axis of the body, hereinafter designated the axial planewave modes, and modes for waves directed at angles to the axis,hereinafter referred to as off-axis modes. In particular, if the onlylight allowed to reflect bidirectionally through the cavity were lightemitted in the axial plane wave modes, so as to effect stimulation ofemission predominately by modeselected plane wave light energy, a highdegree of emissive efficiency would be achieved. The laser output oflight in the plane wave front (the useful portion of the output pulse)would be significantly greater than it is when bidirectional reflectionof light in off-axis modes is allowed to develop in the cavity; the beamspread angle of the output pulse would be reduced; and as a result theoutput intensity, or power per unit solid angle delivered by the laserat any given distance (an inverse function of the beam spread angle),would be advantageously increased. correspondingly, it has been foundthat in general, to the extent that bidirectional reflection of lightemitted in the off-axis modes can be inhibited, the intensity in asingle mode of the laser output pulse may be desirably improved.

In one preferred system for effecting such mode-selective laseroperation, light emitted from the laser body and reflecting back andforth within the cavity is focused as by a suitable lens through a focalpoint intermediate the body and one of the reflective cavity ends. Amask defining an aperture is positioned in the cavity so that theaperture coincides with this focal point. The aperture permits light inselected modes to pass through the focal point, while the surroundingmask, oceluding a portion of the image formed by the lens at the focalpoint, dissipates light energy emitted in other modes. Bidirectionalreflection of light in the cavity is thereby limited to modes for wavesdirected through the aperture by the lens; light emitted in other modescannot pass beyond the focal point to the aforementioned cavity end, andthus cannot reflect bidirectionally between this end and the opposedcavity end, because it is blocked by the mask.

The mask referred to above may be a plane opaque member having a surfaceof minimal reflectivity by its deflection capability for incident lightand it is pierced by a slit, aperture, or other opening of appropriateconfiguration (ordinarily smaller, at least in minimum dimension, thanthe image formed at the focal point by the lens with a conventionallaser as the source); light not directed through the aperture isdeflected by the mask in the region adjacent the aperture.

The flash tube, as it is called, for pumping or providing the initialenergy inversion is usually in the form of a helix concentricallysurrounding and in spatial relationship to the laser core, whose coilsare equally spaced along the length of the laser rod to distribute itsheat emission evenly along the rod. However, it may be seen that theradial heat distribution is quite uneven, the flash tube causing hightemperatures at the periphery of the rod and lower temperatures at theaxis. The thermal stress distribution in the rod is therefore similarlyuneven, causing a change in index of refraction during pumping by theflash tube and a resulting reduction in beam definition, which isintimately related to desired laser output.

Furthermore, stress birefringence is caused throughout the laser rod byuneven temperature, except at its axis, so that light polarizedtangentially will encounter a different index of refraction than lightpolarized radially at all points not on the axis. The total result ofthe varying indices of refraction is a difference in path length withdistance from the axis, and with polarization, and a consequentreduction in beam definition.

At the present time the laser art is proceeding without compensation forthese problems. Lens compensation is impractical, since a fixed lensbecomes useless in the face of a constantly changing variation in theindices of refraction, and, obviously, a series of insertable lenses arealso unsatisfactory.

Accordingly, a primary object of the present invention is to compensatefor the effects of thermal gradients produced in a laser rod by the heatfrom the flash tube.

A more specific object is to control the composition of a laser rod andthe path of resonant light therewithin so that it is not affected by theaforementioned thermal gradients and stress birefringence.

A further specific object is to select a mode of propagation whoseoutput is plane polarized and in a wave front of uniform intensity anduniformly perpendicular to the laser axis, this to be accomplished bytaking the average of the radial and tangential vectors of that mode.

These and other objects are accomplished in one illustrative embodimentof the invention, which features a pin hole in a deflecting plate and alens system for selection of the HE mode of propagation, and aone-eighth wave Faraday rotator for averaging the path .length oftangentially and radially polarized light waves.

Other objects, features, embodiments and modifications are contemplatedand will be described and apparent from the following more detaileddiscussion and reference to the accompanying drawing, wherein:

FIG. 1 diagrammatically represents transverse components of the electricvectors for the four lowest order sets of modes in a dielectricwaveguide and in a laser structure; FIG. 2 is a plot of radiationpatterns or far field distributions of light intensity emitted by alaser operating in each of the modes of FIG. 1;

FIG. 3 is a representation of a typical laser rod;

FIG. 3A is a sectional diagrammatic representation of the compressionand tension on an isolated portion of a plane of the mediumperpendicular to the axis of the laser rod of FIG. 3, through whichlight is propagating stresses being caused by the variation intemperature within the laser rod;

FIG. 4 is a plot of the stress distributions for the tangential andradial directions of polarization;

FIG. 5 is a representation of the laser rod of FIG. 3 shown with theequiphase surfaces or wave fronts, depicting also the variation in indexof refraction due to temperature, radial stresses, and tangentialstresses and the average variation of the index of refraction fortangential and radial stresses attained by the apparatus of theinvention;

FIG. 6 is a schematic representation, partially in isometric, of a laserapparatus constructed according to the principles of the presentinvention, with FIG. 6A being an enlarged view of a portion thereof;

FIG. 6B is a representation of the HE mode of propagation with electricvector directions shown before being rotated and after two one-eighthwave rotators;

FIG. 7 is a plot of the thermal coefficient 7 versus L/diam.configuration constants useful in choosing a host composition desirablefor laser operation according to the invention; and

FIG. 8 shows a table of the values obtained, which are useful inplotting the curve of FIG. 7.

Since the stress components in the radial direction differ from thestress components in the tangential direction, except at the center oraxis of the laser rod, a plane wave of arbitrary polarization willnecessarily be distorted, when passing through the laser, by stressbirefringence produced largely by the flash tube. It will be shown thatit is possible to choose the coefficient of linear expansion, a and thethermal coefficient of the index of refraction, a so that the pathlength of a ray through the edge of the rod is equal to the path lengthof a ray through the edge of the rod, with polarization both in thetangential and radial directions. The single mode I-IE is excited and apin hole of preselected diameter is provided in a deflector between twolenses for isolation of that mode. Also, a Faraday rotator is placed atthe opposite end of the laser rod from the lens-deflector system, foraveraging tangential and radial electric vectors of the I-IE, mode. Astress analysis for extreme laser rod configurations will then bepresented and a plot derived for relating various configurations to thethermal coefficient 7 (related to a L and a from which plot thecomposition of the laser material can be specified for the average ofpolarization in the radial and tangential directions.

Referring first to FIG. 1, the electric vectors for four sets of modesare shown. The first set includes only the HE mode; the second setincludes TE TM 21, TM, HE and TE HE the third set includes EH HE EI-I,HE and EH I-IE and the fourth set includes only the HE mode.

FIG. 2 is a plot of the relative intensity versus u for the four sets ofmodes shown in FIG. 1. This plot essentially represents a Fouriertransformation of the so-called direct image intensity distribution. Bychoosing the proper u, discrimination against certain sets of modes isobtained. For example, if u is chosen to be 3.5, which represents theratio of the circumference of the laser aperture times the sin 6, to thewavelength of a laser ray (the angle 0, will be defined subsequently),sets 2, 3 and 4 of the modes shown in FIG. I will be substantiallyeliminated, since a significant portion of their intensity will be cutoff.

FIG. 3 is a representation of a typical laser rod, that FIGURE being aside view of a substantially cylindrical configuration.

FIG. 3A is a sectional diagrammatic representation of the compressionand tension of an isolated portion of a plane of the mediumperpendicular to the axis of the laser rod of FIG. 3, through which thelight is propagating with the stresses being caused by the variation intemperature within the laser rod, which are, in turn, caused by thenonuniform distribution of the heat from the flash tube as described inthe introduction. The radial stresses and 102 are tension stresses,whether or not the laser rod is analyzed at its center or periphery, asshown by the plot of FIG. 4. For tangential stresses 104 and 106, it maybe seen from the plot of FIG. 4 that these stresses are compressive nearthe periphery of the laser rod and in tension near the axis or center ofthe rod. Relating the aforementioned stresses to indices of refraction,it is known that a medium under tension has a lesser index of refractionthan light being compressed, so that the indices of refraction n willvary approximately according to the plot for stresses shown in FIG. 4.

Referring then to FIG. 5, the approximate variations of the wave fronts(related to the variations of the index of refraction n) are shown aslight emerges from the end of a laser rod. An entering plane wave 107and an optically perfect laser rod are assumed, so that withtangentially polarized light, the variation 108 of the wave front due tostresses is shown for the direction of propagation 110, which isdepicted as a solid arrow. Since the index of refraction is less andconsequently the phase velocity is greater at the axis of the rod thanat its periphery, the equiphase surface of wave front 108 will beconvex, protruding in the direction of propagation near the axis of therod. On the other hand, the variation 114 of the wave front due tostress for radially polarized light is less protruded at the center ofthe rod even though for such polarization the index of refraction isstill less at the center than at the periphery of the rod. Thedifference in convexity between variations 108 and 114 is bestillustrated by reference to FIG. 4 where the index of refraction nincreases downwardly in that plot and for tangential polarizationincreases approximately three times as much as for radial polarization,with the axis of the laser rod as a reference. ln other words, the indexof refraction increases towards the periphery of the rod by arepresentative amount z for radial polarization, whereas for tangentialpolarization, the index of refraction increases by a representativeamount 32.

FIG. 5 also depicts variation 116 of the wave front due to temperature.Since temperature is greatest at the laser rod periphery due to theproximity of the flash tube, it is assumed that the higher index ofrefraction will be at the axis of the rod, since heat has a lesseningeffect on the index of refraction by dispersing the molecules of themedium more as a temperature is increased. This statement is made withthe understanding that index of refraction can also increase withtemperature, depending on the composition of the glass, but thesubsequent analysis and choice of glass composition is made so thatindex of refraction varies inversely with temperature in the mannerassumed. The curvature of the wave front resulting from the variation ofindex of refraction 116 due to temperature will, therefore, be in theopposite direction from its variations 108 and 114 caused by stresses,if the direction of propagation is assumed to be that shown by arrow110.

With the stress distributions and variations in index of refraction asshown by FIG. 4 and as thus described, the apparatus according to thepresent invention is designed to provide a plane polarized output of thelaser rod with a uniform intensity at the output aperture. Means areprovided whereby, if light propagates through a medium having avariation in the index of refraction such as to give the wave frontsrepresented by the curves 108, 114 and 116, is reflected by mirror 118,it is returned to the laser rod in a direction of propagation 112 withpolarization in the radial direction if the polarization in direction110 was tangential, and with tangential polarization if the polarizationin the direction 110 was radial; the variation 120 in index ofrefraction will result, since at any given point such as 122 the samelight wave will encounter an average of the variation in index ofrefraction for tangentially and radially polarized light. It doesnotmatter that the radially polarized light is propagating in one directionand the tangentially polarized light is propagating in the otherdirection, since it is important only to have a plane wave front in aplane through a particular point on the laser axis. Average wave front120 then counterbalances wave front 1 l6 and the result is a plane wave121 with a wave front in a plane perpendicular to the axis.

FIGS. 6 and 6A show the apparatus according to the present invention,wherein a laser rod 12 is concentrically surrounded by a spatiallyrelated helical flash tube 14, the coils of which are sufficientlyproximate to assure uniformity of illumination along the length of thecore 12 and to prevent a significant temperature gradient along thelength of the core. (The energizing circuits are not shown since theyare conventional). An efiicient reflector 118 is adjacent one end of thelaser rod 12 and both ends of the rod are provided with nonreflectivesurfaces 21), with mirror 21 forming the other end of the resonantcavity. As is understood, mirror 21 is partially transmissive so thatthe laser output may escape from the cavity. Pin hole deflector 22 islocated near the output side of the laser rod between two lenses 24 and26, and at the back side of line 24, at a distance equal to the focallens f of that lens. Lens 26 is located on the opposite side of the pinhole deflector and at a distance from lens 24 equal to the total of thefocal lengths f and f of the two lenses. A Faraday rotator oreighth-wave plate 124 is placed at the end of the laser rod nearest tothe total reflector 118.

it should be understood that the description of a helical flash tubesurrounding the laser rod is illustrative only and it is possible tohave a flash tube that is similar in shape to the laser rod and disposedparallel with it to similarly provide uniform heating along the lengthof the rod but uneven heating radially throughout the rod.

The enlarged view, FIG. 6A, is of a portion of the apparatus of FIG. 6,which comprises lens 24 and pinhole absorber 22. Rays propagated alongthe axis 34 from the laser rod are bent on angle 6, with the horizontaland this is the limiting angle along which a ray of laser output lightcan be emitted through the pinhole 23. The actual diameter d of thepinhole 23 is computed by use of the following relationships after u, aparameter related primarily to angle has been chosen, to discriminateagainst certain of the modes of FIG. 1:

sin 0, Ala/21m,

where A is the wavelength of the laser output and a is the radius of thelaser rod 12 d/2 =f tan 6,.

Therefore, by the proper choice of u, and the resulting diameter of thepinhole 23, mode Il-lE is chosen and will be the only mode propagatedwithin the resonant cavity. That mode contains, as shown in FIG. 613 bythe solid arrows, electric vectors 126 in a vertical direction only,with the designation l-lE representing the hybrid electric modes ofpropagation and the first subscript representing; the number of fullperiod variations of the radial component of field along angularcoordinates, and the second subscript defining the number of half periodvariations of the angular component of field along radial coordinates.The center arrow is an electric vector that is equivalent to radialpolarization, whereas the outermost arrows 126 represent electricvectors that are equivalent to tangential polarization. In this way, thepropagation of the HE mode produces varying wave front shapes, accordingto their position in FIG. 68, as was shown by variations 108 and 114 inF IG. 5.

As was previously described, the varying wave front configurations,including that caused by the temperature only and those caused bystress, can be compensated by rotating the electric vectors of FIG. 6Bby so that at a particular point between the laser rod 12 and thereflecting surface 118, the polarization of light will be equally radialand tangential, thereby producing an average effect of the two planes ofpolarization. In order to accomplish this, a Faraday rotator 124, whichis a one-eighth wave rotator, is provided to thereby produce a 45rotation in the light for each of its passages through the rotator.

A Faraday rotator is operated by a magnetic field (not shown) with linesof force parallel to the axis that breaks incident light into twocounterrotating circularly polarized components which then propagatethrough the rotator with different indices of refraction. Thus, thecomponents do not retain a fixed phase relation between themselves asthey pass through the rotator, and upon emerging from the exit end willrecombine to form polarized light of the same character as that whichinitially entered, but the polarization of this light will be rotated byan angular amount which depends upon the Verdet constant of the rotatormaterial, the applied magnetic field, and the axial length of therotator element. By properly choosing a Faraday rotator system accordingto the above, the rotation of 45 of the light in each direction of itspassage can be attained.

By way of example, if light ray 32 comes out of the laser rod and intothe Faraday rotator 124 with polarization in a direction represented byelectric vector 125, that vector will be rotated 45 to a position shownby vector 127 after it emerges from the rotator. The ray will bereflected from element 118 and be directed back towards the laser rod(this reflected light path is depicted by dotted line 33), and uponpassing through the rotator 124 a second time will be rotated another 45and emerge with a direction of polarization represented by vector 129,which is a total of 90 different than the position of entering vector125. The variation in the wave front for the average of the tangentiallypolarized (129) and radially polarized (125) light ray is then as shownby dotted representation 120 of FIG. 5. This variation of wave front isequal and opposite to the variation 116 caused by temperature only, andthe result 121 is therefore a plane wave front with no variation fromaxis to periphery of the laser rod, thereby solving the stressbirefringence problem alluded to previously. As this point, it is onlynecessary to compute the value of thermal coefficient 7 in order tochoose the proper composition of the laser material to satisfactorilyreduce the path difference from axis to periphery to zero. Furthermore,pinhole 23 has eliminated the nonaxial modes and the laser outputintensity is thereby enhanced as described in the introduction.

To compute the range of values required for the thermal coefficient ofthe index of refraction a and the thermal coefficient of linearexpansion 01 in order to minimize the thermal changes in the cavity,calculations will be made for the extreme cases of the rod being verylong, in which case the end effects are small the the circumstancereasonable approximates that of plane stress; and the length being muchshorter than the diameter, in which case the end effects predominate andthe constraints correspond closely to plane stress.

Given a laser of length L and index of refraction n which consists of alarge number of thin concentric cylinders, a temperature gradient isassumed to exist between the center cylinder and the outermost cylinder;however, each cylinder is at a uniform temperature. Remote endreflectors are used which are rigidly mounted and spaced apart bydistance Y. The total path length P in the cavity for a typical ray isgiven y P=nL+Y-L(nl) L+Y. (1) With a difference in temperature betweenthe center and edge of the laser of AT, the difference in optical pathlength for rays which go through the center versus those passing nearthe edge is given by the following equation:

AP=dP/dT AT=dnldT L+(n-l dL/dT AT. The condition that AP be equal to isthen For convenience in later use the parameter n(a,,-+-a, )/a, isdefined at y.

Thermoelastic properties of solid cylinders of radius a and length L arein general complicated problems. Some simplification is introduced byassuming symmetrical heating, so that the temperature Tis a functiononly of the radius.

The stress and strain distributions are easily managed for the limitingcases of L much smaller than a or L much larger than a. The former caseis than of the plane stress, the latter corresponds to plane strain.

Because of the symmetrical heating of the right circular cylinder allthe shear components of stress and strain can be dropped. With acylindrical coordinate system r, 0, and z, the strain components e arerelated to the stress components a by If the rod length L is much largerthan the radius a it is not unreasonable to neglect end effects andconsider only a typical section well removed from the end of the rod.This becomes a case of plane strain, and in such a case two situationsmust be distinguished: if the ends of the rod remain fixed, so that thetotal length does not change, this corresponds to the condition e,,=0.(12) however, if the ends are free of traction so that they can move inresponse to the heating the condition which applies is cargo-"7H7 89 Thecase of interest in a long laser rod is the latter. For it, the stressesare given by "LE 1 a 1 r (A fiifllmna f Trdr- T) The relative importanceof end effects depends on the length to diameter ratio of the rod. Ifthe rod is very long the end efi'ects are small and the problemreasonably approximates that of plane strain. On the other hand, if thelength is much shorter than the diameter the end effects predominate andthe constraints correspond fairly closely to plane stress. If the lengthand diameter are comparable, an intermediate situation between planestress and plane strain applies. To show the range of values requiredfor a and u in order to minimize the thermal changes in the cavity,calculations will be made for these two extreme cases.

The long rod is a commonly used geometry. The case of a large number ofdiscs stacked together would follow plane stress in its thermoelasticbehavior.

Let the laser be in the form of a long cylinder of radius a and length Lwith rigidly fixed, removed end mirrors spaced apart a distance D. Thetwo cases of a solid laser rod or one consisting of many discs will betreated separately. For the solid rod the total optical path lengthP,,(r) for a typical ray parallel to the axis and displaced a distance rfrom the center and with its plane of polarization in the radialdirection is given by where T is the difference in temperature betweenthe center and a point at a distance r from the center. The quantities Pand O are the stress-optic coefficients which relate the change in indexof refraction due to the strains in the directions perpendicular andparallel, respectively to the plane of polarization of the light.Similarly, for the solid rod the total optical path length P 66 (r) fora typical ray parallel to the axis and displaced a distance r from thecenter and with its plane of polarization in the tangential direction isgiven by P900) i n i z:'i" rr) +Q iii h a The strains e e and e M ineqs. (17) and 18) are given in terms of the stresses 0' 5" and cr 90 byeqs. (6), (7) and (8) but with the thermal terms aT dropped from theseequations. The stresses are then related to the temperature distributionby eqs. (l4), (l5), and 16).

The corresponding expressions for the total optical paths for thestacked discs laser configuration are 9 ,JW 7L 1 For the optical pathlengths to be equal for rays through the P MU) n center and at theradius r for the various cases considered above the quantities in thecurly brackets above must be equal {P(e +e +Q600}]}+DL to zero. Thisleads to the various following conditions on the 5 parameters describingthe properties of the glass such as a a, (20) etc.

Long Rod The primes on P',,(r) and P' (r) serve to distinguish thestacked discs case from that of'the solid cylindrical rod. The

a R strains e e and e W in eqs. (19) and (20) are given in terms 1T(1+)(P*Q)2(l P+2VQ1 of the stresses a o and 0- by eqs. (6), (7) and (8)but with the thermal terms aT dropped from these equations. The astresses are then related to the temperature distribution by APw= zn 1 vBefore proceeding to calculate the optical path lengths for 1 1 thevarious cases, a word first on the birefringence produced T Q) V) (1 bythe radial and tangential stresses. Since stress in the radial (29)direction, 0' differs from stress in the tangential direction, a exceptat the center of the rod a plane wave of arbitrary w flg I: +Qpolarization will of necessity be distorted on passing through 2 1 2 2the laser. It is possible to choose the expansion coefficient and (30)the thermal coefficient of index in such a way as to make the path for aray through the center of the rod be equal to the Stacked DISCS pathlength of a ray through the edge of the rod with polariza- 1 tiontangential and radial, but in general not for both. If the AP T raythrough the center has an optical path equal to that of an edge raywhich is tangentially and radially polarized, one can 4:50. Q) 12+ ,0]excite a single lowest order l-lE mode. T

Carrying out the indicated substitutions the differences in (31) opticalpath lengths, AP,,(r), Ap (r), AP,,.(r), and AP' (r), for rays at thecenter and at the distance r from the center for y n l R the four casesconsidered above become AP ("V-0 an 11 4: T 1+ V) (P Q) 2VP Q AP-(T)=1LL{T0l 35 APIH+APIM I 0 a To: (1 Q)+T( Q)]} +a P +Q APBB( Tl/L{T n1i 40 (33) To obtain a rough indication of the values that are typicalLong Rod pear in the plot of FIG. 7. A number of laser glasses andcommercially available glass K271 mp it w hav bee r saskr 5 w f fi lfiif li ii ders, the long rod, and the stacked discs, and the results ap- 1=nLT{a 'n aa[P )-|Q(- In an actual laser, bychoosing the lengths of therod properly and with careful attention given to means for avoiding thecomplications involved in stress induced birefringence, the valuesrequired for a and -y are intermediate the extremes indicated in FIG. 8.The measured values straddle the values shown in FIG. 8; hence it ispossible to design thermally stable laser cavities by the choice of aparticular mode, having both radial and tangential components, andaccordingly choosing -y and then L/diam. values from FIG. 7. If the HEmode is isolated by the apparatus of FIGS. 6 and 6A, values of 0.88 fory and approximately 1.10 for the length to diameter ratio satisfy therequirement of compensating for the thermal distribution problemdescribed. However, these values are for illustrative purposes only. Itis sometimes desirable to choose the thermal coefficient 7 anywhereintermediate the extremes shown on the plot of FIG. 7, according to theavailable laser material. In

this way, it is also possible to design the composition of lasermaterial according to the length-to-diameter ratio desired which isusually the greatest L/diam. ratio possible within the extremes shown.For instance, a 'y of approximately 0.93 could be selected. Also, by notchoosing the exact average of radial and tangential polarization, thethermal gradient effects are nevertheless substantially eliminated,although the stress birefringence problem is not as satisfactorilysolved.

In any embodiment of the invention, it is necessary that the glass basehave an a which is negative in order that algebraic.

cancellation of the other positive effects of thermal gradient? ispossible. In the present invention, the laser device has a mode limitingmeans which limits the mode being propagated to only the l-IE mode.Since the l-IE mode has both radially polarized vectors and tangentiallypolarized vectors, a polarization rotator is included so that alternatepasses of light are successively radially and tangentially polarized toprovide an optical path length equal to the average of the radial andtangential polarized path length. The composition of the materialthrough which the laser light is propagated is chosen so that thecoefficient of linear expansion, the temperature coefficient of theindex of refraction, Poissons ratio, and the stress-optical effectequalized the optical path length for a ray through the center of thebody with the average value of the optical path length for radially andtangentially polarized light at the edge of the laser body. With othervariables remaining constant, any glass with a negative a, will be morethermally stable than the glass with a positive a,, for the averagevalue of the optical path length for radially and tangentially polarizedlight. However, any glass that has a negative coefiicient of the indexof refraction of a, can be optimized by well-known recognized method ofcomputer regression analysis. Thus, a computer can be programmed so thatthe constituents in the glass that tend to make a negative are revealed,along with those constituents that tend to make the linear expansioncoefficient, Poissons ratio and the stress-optical coefiicient positive.In short, the invention resides in the laser device with a mode limitorfor mode limiting those modes which have both radially and tangentiallypolarized vectors and to propagate these modes in a glass host with an asufficiently negative to algebraically cancel the other positive factorseffected by thermal gradients. The device also contains a rotator forrotating alternate passes of light 90 so that those vectors which wereradially polarized in one pass become tangentially polarized in asubsequent pass. Thus, providing an optical path length equal to theaverage value for radial and tangential polarization.

The following glass compositions have been found to have a coefficientof linear expansion, a temperature coefficient of the index ofrefraction, Poissonratio, and a stress-optical effect such that thedifference between the optical path length for a ray at the center ofthe body and the average value of the m s l P h. l thfqt test s!!!9.taas ntia lx latias Us i t new," V.,..? ,,7 passing along the bodynear the periphery thereof is essentially zero when a temperaturegradient is present within the body from the center to the periphery:

It is also possible to select a thermally stable glass base for ionsother than M 0 For a consideration of just the glass base for other Nap,concentrations or other active ions all components in the above examplescan be multiplied by fact of 1.01 thus giving a base for the glasswithout an active ion. This can be considered as the glass into whichother ions can be doped in small amounts without efiecting the thermalproperties of the glass. The above examples represent a preferredembodiment. Other ranges are possible and are contemplated as part ofthe invention.

While a particular embodiment and analysis useful for the generalproblem have been discussed, other embodiments and configurations arecontemplated. The invention is not to be limited by such discussions butis to be accorded the full scope of the following claims.

I claim:

1. A laser apparatus including a laser body in an optically regenerativecavity having an optical axis within said cavity and containing amode-limiting means including an aperture along said axis fordiscriminating against all but the HE mode, and a polarization rotatoralong said axis is provided for rotating the polarization of the output90 whereby alternate passes of light in paths along the length of saidaxis are successively radially and tangentially polarized to provide anoptical path length equal to the average of the radial and tangentialpolarized light path lengths, the composition of the material of saidbody being so chosen that the coefficient of linear expansion, thetemperature coefficient of the index of linear expansion, thetemperature coefficient of the index of refraction,

Poissons ratio, and the stress-optical effect essentially render zero,the difference between the optical path length for a ray through thecenter of the body and the average value of the optical path length forradially and tangentially polarized light passing along said body nearthe periphery thereof when a temperature gradient is present within thebody from its center to its periphery.

2. The apparatus according to claim 1 wherein the polarization rotatoris a one-eighth wave Faraday rotator.

3. The apparatus according to claim 2 wherein the mode limiting meansalso includes two parallel lenses arranged adjacent to one end of thelaser body, the lenses being spaced from each other a distance equal tothe sum of their focal length and disposed coaxially within the laserbody and the mode-selecting aperture being provided in a diaphragmarranged parallel to and between the lenses at the common focal point.

4. The apparatus according to claim 3 wherein said aperture is placedbetween a totally reflecting mirror and a partially reflecting mirrordisposed respectively one at each end of said laser body to provide saidoptical regenerative cavity.

5. A laser device including a body of solid laser material in anoptically regenerative cavity having an optical axis within said cavity,the composition of the material of said body being so chosen that thecoefficient of linear expansion, the temperature coefficient of the index of refraction, Poisson's ratio and the stress-optical effectessentially render zero, the difference between the optical path lengthsfor a ray through the center of the body and the average optical pathlength for radial and tangential polarized light passing along said bodynear the periphery thereof when a temperature gradient is present withinthe body from its center to its periphery, said body having the shape ofa right, circular, cylindrical rod with end faces and a cylindricalsurface;

a mode-limiting means including an aperture along said axis fordiscriminating against all but the HE mode;

a pair of parallel lenses disposed near one end of said body in planesperpendicular to the axis of said body, said lenses being separated bythe sum of their focal lengths and being coaxial within said body;

said aperture being defined by an opening in a light opaque plate, saidplate being disposed parallel to and between said lenses and at the backfocal point of the lens nearest said body, said plate having a pinholeat its center forming said aperture, said pinhole having a diameter dbearing the following relationship to the focal length F, of the nearestlens d/2#, tan 9, and =u)\/21ra for small angles, where A is thewavelength of the laser output, a is the radius of said laser body, u isa parameter related primarily to angle and is chosen to eliminate allbut the HE, mode of propagation of said laser output, which mode hasboth radial and tangential electric vectors and whose intensity andphase are constituted such as to give approximately a uniform planepolarized distribution across the aperture of the rod and which is thelowest order hybrid mode; and

a magneto-optic rotator disposed adjacent the other end of said laserbody for averaging the radial and tangential vectors of said mode.

6. The apparatus according to claim 5 wherein substantially totallyreflecting mirror is disposed in a plane perpendicular to said bodyaxis, coaxial with said body, adjacent said magnetooptic rotator andfurther from said body than said rotator for receiving rays from saidrotator and retransmitting the rays to said rotator, with theretransmitted rays being rotated by said rotator one-eighth wave fromtheir direction before passing through said rotator; and

a partial reflecting mirror is coaxial with said body adjacent the otherend of said body from. said rotator and further away from said body thansaid lenses, said partially reflecting mirror defining with said totallyreflecting mirror a laser resonant cavity. 7. The laser apparatusaccording to claim 6 wherein said body is formed of glass, said glassconsisting essentially of the following approximate composition inweight percent:

8. The laser apparatus according to claim 6 wherein said body is fonnedof glass, said glass consisting essentially of the following approximatecomposition in weight percent:

LiO; 40.0 N3 0 B30 20.0 1.3,0 22.0 M1 0, 1.0

9. The laser apparatus according to claim 6 wherein said body is formedof glass, said glass consisting essentially of the following approximatecomposition in weight percent:

Lio, 420 K20 23. 1 Ba() I 8.8 ZnO 4.0 A1 0,, 2.5 w o, 5.9 si ,o 2.7 Nap,1.0

2. The apparatus according to claim 1 wherein the polarization rotatoris a one-eighth wave Faraday rotator.
 3. The apparatus according toclaim 2 wherein the mode limiting means also includes two parallellenses arranged adjacent to one end of the laser body, the lenses beingspaced from each other a distance equal to the sum of their focal lengthand disposed coaxially within the laser body and the mode-selectingaperture being provided in a diaphragm arranged parallel to and betweenthe lenses at the common focal point.
 4. The apparatus according toclaim 3 wherein said aperture is placed between a totally reflectingmirror and a partially reflecting mirror disposed respectively one ateach end of said laser body to provide said optical regenerative cavity.5. A laser Device including a body of solid laser material in anoptically regenerative cavity having an optical axis within said cavity,the composition of the material of said body being so chosen that thecoefficient of linear expansion, the temperature coefficient of theindex of refraction, Poisson''s ratio and the stress-optical effectessentially render zero, the difference between the optical path lengthsfor a ray through the center of the body and the average optical pathlength for radial and tangential polarized light passing along said bodynear the periphery thereof when a temperature gradient is present withinthe body from its center to its periphery, said body having the shape ofa right, circular, cylindrical rod with end faces and a cylindricalsurface; a mode-limiting means including an aperture along said axis fordiscriminating against all but the HE11 mode; a pair of parallel lensesdisposed near one end of said body in planes perpendicular to the axisof said body, said lenses being separated by the sum of their focallengths and being coaxial within said body; said aperture being definedby an opening in a light opaque plate, said plate being disposedparallel to and between said lenses and at the back focal point of thelens nearest said body, said plate having a pinhole at its centerforming said aperture, said pinhole having a diameter d bearing thefollowing relationship to the focal length F1 of the nearest lens d/2 f1tan theta 1 and theta u lambda /2 pi a for small angles, where lambda isthe wavelength of the laser output, a is the radius of said laser body,u is a parameter related primarily to angle and is chosen to eliminateall but the HE11 mode of propagation of said laser output, which modehas both radial and tangential electric vectors and whose intensity andphase are constituted such as to give approximately a uniform planepolarized distribution across the aperture of the rod and which is thelowest order hybrid mode; and a magneto-optic rotator disposed adjacentthe other end of said laser body for averaging the radial and tangentialvectors of said mode.
 6. The apparatus according to claim 5 whereinsubstantially totally reflecting mirror is disposed in a planeperpendicular to said body axis, coaxial with said body, adjacent saidmagneto-optic rotator and further from said body than said rotator forreceiving rays from said rotator and retransmitting the rays to saidrotator, with the retransmitted rays being rotated by said rotatorone-eighth wave from their direction before passing through saidrotator; and a partial reflecting mirror is coaxial with said bodyadjacent the other end of said body from said rotator and further awayfrom said body than said lenses, said partially reflecting mirrordefining with said totally reflecting mirror a laser resonant cavity. 7.The laser apparatus according to claim 6 wherein said body is formed ofglass, said glass consisting essentially of the following approximatecomposition in weight percent: LiO2 41.6 Na2O 14.4 BaO 26.7 Nb2O3 15.4Sb2O3 0.9 Nd2O3 1.0
 8. The laser apparatus according to claim 6 whereinsaid body is formed of glass, said glass consisting essentially of thefollowing approximate composition in weight percent: LiO2 40.0 Na2O 7.0BaO 20.0 La2O3 22.0 Nd2O3 1.0
 9. The laser apparatus according to claim6 wherein said body is formed of glass, said glass consistingessentially of the following approximate composition in weight percent:LiO2 42.0 K2O 23.1 BaO 18.8 ZnO 4.0 Al2O3 2.5 Nb2O3 5.9 Sb2O3 2.7 Nd2O31.0